Set of 3 Pythagorean triangles equal perimeters and area in A.P.

 
 

Here are sets of 3 Pythagorean triangles with equal perimeters and area in arithmetic progression.

Note that the perimeters are multiples of 120.

pt-area-in-ap-1

 
  
The pattern breaks in this example:

The perimeter is not a multiple of 120.

 
3-pt-in-ap-1
 
 

Find other sets of 3 Pythagorean triangles with equal perimeters that are not multiples of 120.

 

                                                 ………………..   #2   ………………..                             

 

multiples of 120 appear again :

Notice that the radius of incircle of the first triple in each set is a multiple of 120

pt-area-in-ap-2

 

                                                 ………………..   #3   ………………..                             

 
No multiples of 120 :

pt-area-in-ap-3

 

                                                 ………………..   #4   ………………..                             

 

One triple is primitive in each set:

pt-area-in-ap-4

 
Find other such sets – that include a primitive – as shown in #4

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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2 Responses to Set of 3 Pythagorean triangles equal perimeters and area in A.P.

  1. Paul says:

    Here are a few sets

    {315,624,699,1638}
    {364,585,689,1638}
    {455,504,679,1638}
    Areas are 98280 & 106470 & 114660 AP is 8190

    {3150,6240,6990,16380}
    {3640,5850,6890,16380}
    {4550,5040,6790,16380}
    Areas are 9828000 & 10647000 & 11466000 AP is 819000

    {3465,6864,7689,18018}
    {4004,6435,7579,18018}
    {5005,5544,7469,18018}
    Areas are 11891880 & 12882870 & 13873860 AP is 990990

    {630,1248,1398,3276}
    {728,1170,1378,3276}
    {910,1008,1358,3276}
    Areas are 393120 & 425880 & 458640 AP is 32760

    {1575,3120,3495,8190}
    {1820,2925,3445,8190}
    {2275,2520,3395,8190}
    Areas are 2457000 & 2661750 & 2866500 AP is 204750

    {2730,10472,10822,24024}
    {3575,9912,10537,24024}
    {4620,9152,10252,24024}
    Areas are 14294280 & 17717700 & 21141120 AP is 3423420

    {945,1872,2097,4914}
    {1092,1755,2067,4914}
    {1365,1512,2037,4914}
    Areas are 884520 & 958230 & 1031940 AP is 73710

    {1890,3744,4194,9828}
    {2184,3510,4134,9828}
    {2730,3024,4074,9828}
    Areas are 3538080 & 3832920 & 4127760 AP is 294840

    {3825,8004,8871,20700}
    {4500,7475,8725,20700}
    {5796,6325,8579,20700}
    Areas are 15307650 & 16818750 & 18329850 AP is 1511100

    {1260,2496,2796,6552}
    {1456,2340,2756,6552}
    {1820,2016,2716,6552}
    Areas are 1572480 & 1703520 & 1834560 AP is 131040

    {5355,10608,11883,27846}
    {6188,9945,11713,27846}
    {7735,8568,11543,27846}
    Areas are 28402920 & 30769830 & 33136740 AP is 2366910

    {4590,6496,7954,19040}
    {4879,6240,7921,19040}
    {5440,5712,7888,19040}
    Areas are 14908320 & 15222480 & 15536640 AP is 314160

    {5460,20944,21644,48048}
    {7150,19824,21074,48048}
    {9240,18304,20504,48048}
    Areas are 57177120 & 70870800 & 84564480 AP is 13693680

    {3780,7488,8388,19656}
    {4368,7020,8268,19656}
    {5460,6048,8148,19656}
    Areas are 14152320 & 15331680 & 16511040 AP is 1179360

    {4725,9360,10485,24570}
    {5460,8775,10335,24570}
    {6825,7560,10185,24570}
    Areas are 22113000 & 23955750 & 25798500 AP is 1842750

    {2205,4368,4893,11466}
    {2548,4095,4823,11466}
    {3185,3528,4753,11466}
    Areas are 4815720 & 5217030 & 5618340 AP is 401310

    {2520,4992,5592,13104}
    {2912,4680,5512,13104}
    {3640,4032,5432,13104}
    Areas are 6289920 & 6814080 & 7338240 AP is 524160

    {4485,5980,7475,17940}
    {4715,5772,7453,17940}
    {5244,5265,7431,17940}
    Areas are 13410150 & 13607490 & 13804830 AP is 197340

    {2835,5616,6291,14742}
    {3276,5265,6201,14742}
    {4095,4536,6111,14742}
    Areas are 7960680 & 8624070 & 9287460 AP is 663390

    {5670,11232,12582,29484}
    {6552,10530,12402,29484}
    {8190,9072,12222,29484}
    Areas are 31842720 & 34496280 & 37149840 AP is 2653560

    {5040,9984,11184,26208}
    {5824,9360,11024,26208}
    {7280,8064,10864,26208}
    Areas are 25159680 & 27256320 & 29352960 AP is 2096640

    {4410,8736,9786,22932}
    {5096,8190,9646,22932}
    {6370,7056,9506,22932}
    Areas are 19262880 & 20868120 & 22473360 AP is 1605240

    {4095,8112,9087,21294}
    {4732,7605,8957,21294}
    {5915,6552,8827,21294}
    Areas are 16609320 & 17993430 & 19377540 AP is 1384110

    {5985,11856,13281,31122}
    {6916,11115,13091,31122}
    {8645,9576,12901,31122}
    Areas are 35479080 & 38435670 & 41392260 AP is 2956590

    {6615,13104,14679,34398}
    {7644,12285,14469,34398}
    {9555,10584,14259,34398}
    Areas are 43341480 & 46953270 & 50565060 AP is 3611790

    {3927,21060,21423,46410}
    {5005,20400,21005,46410}
    {6188,19635,20587,46410}
    Areas are 41351310 & 51051000 & 60750690 AP is 9699690

    Paul.

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